Supplementary Information: Superconducting spintronic tunnel diode

E. Strambini ,1, ∗ M. Spies ,1, † N. Ligato ,1 S. Ilic ,2 M. Rouco ,2 Carmen González Orellana ,2 Maxim Ilyn ,2 Celia Rogero ,2, 3 F.S. Bergeret ,2, 3 J. S. Moodera ,4 P. Virtanen ,5 T. T. Heikkilä ,5 and F. Giazotto 1, ‡ 1NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56127 Pisa, Italy 2Centro de Física de Materiales (CFM-MPC) Centro Mixto CSIC-UPV/EHU, E-20018 Donostia-San Sebastián, Spain 3Donostia International Physics Center (DIPC), 20018 Donostia–San Sebastián, Spain 4Physics Department and Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 5Department of Physics and Nanoscience Center, University of Jyväskylä, P.O. Box 35 (YFL), FI-40014 University of Jyväskylä, Finland

Besides the superconductor/ferromagnetic insulator/normal metal (S/FI/N) structure shown in the main text, different material combinations with equivalent spin-filtering and spin-splitting have been investigated. Most notably, a FI/S/I/F structure (where I is an insulator and F is a metallic ferromagnet) have been investigated. Differing from S/FI/N junctions, here, the spin-filtering and spin-splitting are decoupled. The former is still provided by the FI/S interface, while the latter is due to the I/F tunnel barrier.
Samples are cross-bars made by electron-beam evaporation employing an in-situ shadow mask on a substrate of fused silica and consist of layers of EuS(14 nm)/ Al(9 nm)/ AlO x (4-5 nm)/ Co(10 nm)/ CaF(7 nm). The overlap between the Al and the Co strip has an area of 300 × 300 µm 2 . The tunneling spectroscopy is carried out at cryogenic temperatures down to 50 mK in a filtered cryogen-free dilution refrigerator. The I(V ) characteristics are obtained from DC four-wire measurements as described in the main text.
The data analysis on the I(V ) characteristic at B = 0 is shown in Fig. S2. Notably, as shown in Fig. S2a and b, in this device the zero-bias conductance is more pronounced with respect to the the S/FI/N sample shown in the main text. On the other hand, large spin-splitting and spin-filtering are visible even at zero magnetic field thanks to the stronger ferromagnetism of the EuS layer. Therefore, even if the rectification is smaller with respect to the S/FI/N devices (here the maximum rectification is ∼ 18% as estimated in Figs. S2d and f) the presence of a sizable rectification, even in the absence of an external magnetic field, makes it appealing for applications. Moreover, differing from S/FI/N junctions where the direction of the diode is fixed by the sign of the exchange interactions at the EuS/Al interface, in this typology of device the direction of the diode can be inverted by changing the relative magnetization of the FI and F layers (parallel or anti-parallel), introducing additional functionalities.

A. Resolution and Noise equivalent Power
The sizable rectification of the superconducting tunnel diode observed both in the direct (i) and in the transverse (ii) configuration can find an immediate application in the detection of electromagnetic radiations. Starting from the characterizations presented in the main text, it is possible to estimate the maximum resolution and the noise equivalent power (NEP) of a detector based on this technology. For configuration (i) the DC response to a sinusoidal low-frequency ( ω ∆) AC signal (V AC = V 0 sin (ωt)) can be estimated by averaging the current response over the signal time period T : and the resulting power dissipated by the signal reads: In Here I AC = I 0 sin (ωt) is the AC signal to probe. Here, the resulting NEP evaluated probably due to the small impedance of the tunnel junction favoring the detection in the closed circuit configuration.
In the transverse configuration (ii), the rectification response can be estimated from Eq. S3 in a similar way, starting from the V sym (I Cu ) characteristics shown in Fig. 3 and Fig. 4 of the main text. Differing from configuration (i), the power will be mainly dissipated in the Cu strip and can be estimated from its simple Ohmic response: where R 2Ω is the lateral resistance of the Cu lead at the interface with the EuS.
In Fig. S4a  In Fig. S5 we estimate the resolution and NEP in direct configuration (i) for the additional sample structure FI/S/I/F introduced earlier (I(V ) shown in Fig. S2). Thanks to the strong ferromagnetism of this device even at no applied external magnetic field the NEP reaches an impressive ∼ 10 −18 -10 −19 W/ √ Hz, but only for low powers due to the higher impedance (four orders of magnitude) of the tunnel junction. Such a high impedance improves the NEP in the closed circuit configuration reaching values of ∼ 10 −17 -10 −18 W/ √ Hz, which is much smaller then the N/FI/S counterpart.

B. Intrinsic noise for a rectifier detector
The above values for the Noise Equivalent Power are the amplifier contributions to the total NEP. In good detectors they should be below the intrinsic noise values. Let us consider the case where configuration (i) is used as a direct detector, where the incoming radiation power with frequency ω ∆ is rectified to give the dc current that provides the detector response. Following [1], the dc current in this regime is given by The intrinsic noise is due to the thermal Johnson-Nyquist noise across the junction, with zero-frequency spectral density and using the linear junction conductance G. The intrinsic NEP is obtained from Hereω = ω/∆,h = h/∆, t = k B T /∆ and g = G·50 Ω. Note that this does not include the contribution from non-ideal quantum efficiency.
To model heating effects in the "direct" configuration (i), we apply the thermal model where we include electron-phonon thermal relaxation, the heat flow and Joule heat associated with tunneling, and Joule heat dissipated on the normal side. In the parameter range considered, heat diffusion out of the large-area junction is negligible.
Expressions forQ eph andQ tun can be found in Ref. [2]. The Joule power dissipated on the N-side we approximate with P N, is the lateral resistance of the copper under junction of size L x × W with layer thickness t Cu .
We for simplicity neglect here non-uniformity in the current distribution and temperature in the junction, which is accurate for R T R Cu . Here R T ∼ 2R Cu , which is sufficient for estimating the relative order of magnitude of the effects. The non-uniformity is important for configuration (ii), which we discuss separately below.
The resulting temperatures T N and T S are shown in Fig. S6. There is generally substantial heating on the S side, because most of the power is dissipated as Joule heat in the junction.
Large part of this enters the S side, which has both smaller volume and smaller e-ph coupling than the N side.
The resulting symmetric and antisymmetric components of the IV curve are shown in

B. Lateral configuration (ii).
To obtain input parameters for the modeling including the thermoelectric effects, we have fitted the IV data sets in the lateral configuration (ii) with the following model: where G T , a, V off,j and T N,j are the fit parameters, corresponding to a set of values V i and V H,j for the bias and heating voltages, and dI expt /dV the observed differential conductance.
The lateral resistance is R x = aR H , where R H ≈ 4.2 kΩ is the resistance relating the heating voltage to the heating current, where I N F IS (V, T N , T S ) is the current-voltage relation discussed in Ref. [3]. We include the effects of Γ and other parameters affecting the density of states of the superconductor as in the main text, determined by separate fits done for V H = 0. We assume the order parameter ∆ remains roughly constant in the parameter range considered, in which case the differential conductance is independent of the superconductor temperature T S .
After obtaining the above parameters, we find the temperature T S of the superconducting side by solving the thermal balance model whereQ tun is the tunneling heat current to S obtained analogously as in Eq. (S12) (see Ref. [3]). It is balanced by electron-phonon relaxation, with heat currentQ e−ph as described in Ref. [2], using literature parameters for Aluminum electron-phonon coupling [4], and including the effects from spin splitting, Γ and spin-flip scattering. The resulting T S is shown in Fig. S8.  Based on the temperature difference obtained from the model, we show in Fig. S10 the Seebeck coefficient corresponding to the voltages in Fig. S9 and temperatures in Fig. S8.
These results are all based on subtracting the counterfactual model result including only rectification, and hence the accuracy is limited to providing rough guidance of the likely order of magnitude.
Finally, we can note that the relative strength of the rectification and thermoelectricity varies depending on the junction length L x . If the junction is very short, there is no transverse voltage drop or rectification, whereas if the junction is very long the rectification is large. We can estimate the length scale on which thermoelectricity starts to dominate as follows.
First, from characteristics of I N F IS one can observe the rectification scales with the dimensionless parameter ∼ V /∆ describing the transverse voltage. On the other hand, thermoelectricity scales with ∼ δT /T where δT = T N − T S is the temperature difference.
For thermoelectricity to be large and dominating, we then want to simultaneously have δT /T ∼ 1 and eV ∆. In a rough estimate, under such conditions, the heat balance equation (S13) can be approximated with We assume here that the phonon system is at zero temperature. Here,g is the ratio of suppression factors due to superconductivity, in the tunneling compared to that of e-ph coupling. Moreover, R T = ρ /(L x W ) is the tunneling resistance where L x is the junction length, W its width and ρ the square resistivity, and V S = L x W t S and Σ S are the volume and the electron-phonon coupling in the superconductor [4], and t S is the superconductor thickness. At low temperatures (0.2-1.2 K), based on numerical calculations forQ e−ph /Q tun , we estimateg ≈ 2(k B T /∆) 2 . Moreover, since T is maintained above the phonon temperature by Joule heating, where ρ N , Σ N are the resistivity and e-ph coupling on the normal side, and j H the current density of the heating current. Finally, the condition eV ∆ is equivalent to L x L x,c = ∆ 2g /T 2 e 2 ρ N Σ N T 3 x,S ≈ 2ρ t S Σ S ρ N Σ N ≈ 100 µm . (S18) Note that the precise value depends on material parameters, also because our estimate forg depends somewhat on values of Γ and spin-flip scattering in the superconductor. The result is however well consistent with the fact that in the experiment of the main text, rectification dominates thermoelectricity.